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Analysis of Herzog's LF Phase Compensation Method
for Current-transformers by David Knight.
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Abstract: A Low-frequency current-transformer phase compensation scheme described in US Patent No, 4739515 is examined. The method gives an improvement in phase performance but degrades amplitude performance, leading to unnecessary low-frequency balance-point error in bridge applications. |
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Herzog's LF compensation method: In the design of current-transformer bridges for RF applications, it is conventional to correct for the falling reactance of the current-transformer secondary winding at low frequencies by modifying the voltage sampling network. It appears feasible however, that compensation might be obtained by modifying the current transformer loading network; particularly by loading the transformer with two resistors in parallel and placing a capacitor in series with one of those resistors. As the reactance of the capacitor increases with diminishing frequency, the current-transformer load impedance rises, thereby increasing the output voltage. This 'unloading' method was patented by Will Herzog, K2LB, in 1988 [US Pat. 4739515], and further background information is given in ref. [1]. How to choose the component values for the loading network is not a straightforward matter however, and neither of the references offers a design procedure; but here it will be shown that best compensation is achieved when the circuit exhibits a condition known as critically-damped resonance. |
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For the purpose of low-frequency
analysis, we will neglect the propagation delay and secondary
parasitic capacitance of the current transformer. Thus, in the
circuit shown on the right, the current analog Vi is given by: Vi = ( I/Ni )[ Ri // jXLi // (Rh + jXCh) ] and when the generator is loaded with a resistance R0: I = V / R0 and so: Vi = ( V/NiR0 )[ Ri // jXLi // (Rh + jXCh) ] If we define the transfer function as: |
 Fig. 1 |
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hi =
Vi / V then: hi = [ Ri // jXLi // (Rh + jXCh) ] / NiR0 The network impedance inside the square brackets of this expression is cumbersome when expanded, and so we will work with the reciprocal transfer function and treat it as an admittance, i.e.: 1/hi = NiR0 [ (1/Ri) + (1/jXLi) + 1/(Rh + jXCh) ] Multiplying numerator and denominator of the rightmost term by the complex conjugate of its denominator then gives: 1/hi = NiR0 [ (1/Ri) + (1/jXLi) + (Rh - jXCh)/(Rh² + XCh²) ] which, remembering that 1/j=-j, can be separated into real and imaginary parts:
1/h¥ = NiR0 { (1/Ri) + (1/Rh) } from which it can be seen that at infinite frequency, the current transformer load is simply Ri//Rh. Thus, when designing a bridge, it is Ri//Rh, rather than Ri on its own which must be used when determining the ratio of the voltage sampling network impedances. Hence, if a voltage sampling network is to be pre-chosen, we might, for example, decide to impose the condition Ri//Rh=50W. This gives us a link to standard design procedure, but before we can determine the actual resistor values we must explore the conditions under which circuit resonance can occur. The current transformer secondary inductance Li will resonate with the compensation-network capacitance Ch if the imaginary part of equation 1 can go to zero at some finite frequency, i.e., when: (1/XLi) + XCh/(Rh²+XCh²) = 0 which can be expanded and rearranged thus: (1/2pfLi) - 1/[2pfCh (Rh²+XCh²) ] = 0 1/Li = 1/[Ch (Rh²+XCh²) ] Rh²+XCh² = Li/Ch XCh² = (Li/Ch) - Rh² XCh = ±Ö[(Li/Ch) - Rh²] Since capacitive reactance is negative, the negative sign of the square-root is appropriate, hence: 1/(2pf0Ch) = Ö[(Li/Ch) - Rh²]
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In order to evaluate Herzog's LF compensation scheme, we can
imagine that the transformer is used in a bridge which, if it
were to operate ideally, would have a current transfer function
at balance: h¥ = (Ri//Rh) / NiR0 Since it does not operate ideally, it has an actual transfer function: h0 = [ Ri // jXLi // (Rh + jXCh) ] / NiZbal where Zbal is the actual load required in place of R0 in order to balance the bridge. Note that this transfer function is real, and numerically identical to the ideal case transfer function, because the process of adjusting Zbal to balance the bridge will give Zbal the same phase angle as the current transformer secondary network impedance [Ri//jXLi//(Rh+jXCh)]. The ratio of two impedances having the same phase angle is scalar (AC theory, theorem 1-21.8). Hence, equating the ideal and actual transfer functions: (Ri//Rh) / NiR0 = [ Ri // jXLi // (Rh + jXCh) ] / NiZbal which gives:
Zbal = [ R0/(Ri//Rh) ] / { (1/Ri) + [ Rh/(Rh²+XCh²) ] -j[ (1/XLi) + XCh/(Rh²+XCh²) ] } The way in which Zbal changes with frequency characterises the bridge, because the ratio |Zbal|/R0 gives the magnitude error in the balance condition, and the phase of Zbal gives the phase error. Hence, what we need to do now is to obtain expressions for the magnitude and phase of Zbal and calculate these quantities for various values of Ri and Rh. To this end we can write: |
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Zbal = k / (G +jB) where: k = R0/(Ri//Rh) G = (1/Ri) + Rh/(Rh²+XCh²) and -B = (1/XLi) + XCh/(Rh²+XCh²) Hence: Zbal = k (G -jB) / (G² +B²) which gives: |Zbal| = k [Ö(G² +B²)] / (G² +B²) i.e.,
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 Fig. 2 |
(-B/G)| Now, setting up a spreadsheet (Herzog_LF.ods) with columns for Log(f), f, G, -B, |Zbal| and f, we can create graphs and tables which show how |Zbal| and f change with frequency. Shown below in figs. 3 and 4 and table 2 are the results of calculations performed for a current transformer with Li=10mH, and a secondary load at high frequencies (Ri//Rh) of 50W. The characteristic resistance R0 is also 50W, making k =R0/(Ri//Rh)=1. The capacitance Ch is calculated using the critical damping condition (equation 2), and the required values for various combinations of Ri and Rh are given in table 1 below. Also given are the required capacitances when Li=20mH, showing that they scale in proportion to the inductance. It transpires that if the transformer secondary inductance is multiplied by some factor, then the frequency at which a given phase or magnitude error occurs is divided by the same factor. Consequently, the graphs for the 10mH case can be made to serve for the 20mH case (say) by drawing a new frequency scale with all of the numbers divided by 2. |
| Table 1. Compensation capacitance for various choices of Ri and Rh. |
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/ W |
/ W |
( Li=10mH ) |
( Li=20mH ) |
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Uncompensated |
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Maximum LF boost. |
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| Table 2. Magnitude and phase error of Herzog LF-compensated bridges (data used for the graphs in figures 3 and 4, obtained from the spreadsheet calculation Herzog_LF.ods). Li=10mH, R0=50W, Ri//Rh=50W (this model does not include propagation delay). The phases and magnitudes for any other transformer secondary inductance Li' can be obtained by multiplying the entries in the frequency column by Li/Li', (where Li=10mH). The Ch values at the heads of the columns can be multiplied by Li'/Li to find the corresponding capacitance. |
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Rh ®¥ Ch = 0 |
Rh =100W Ch = 1nF |
Rh = 75W Ch=1778pF |
Rh = 60W Ch=2778pF |
Rh = 52W Ch=3698pF |
Rh = 50W Ch = 4nF |
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| 1.00 | 39.1 | 38.5 | 58.2 | 41.7 | 65.2 | 37.6 | 70.2 | 32.2 | 72.5 | 28.0 | 72.9 | 26.7 |
| 1.12 | 40.8 | 35.3 | 61.2 | 35.4 | 66.9 | 30.2 | 69.9 | 24.6 | 70.7 | 20.7 | 70.8 | 19.6 |
| 1.26 | 42.3 | 32.3 | 63.0 | 29.3 | 66.9 | 23.6 | 68.2 | 18.3 | 68.0 | 15.0 | 67.8 | 14.2 |
| 1.41 | 43.6 | 29.4 | 63.6 | 23.6 | 65.8 | 18.0 | 65.8 | 13.5 | 65.1 | 10.8 | 64.8 | 10.1 |
| 1.58 | 44.7 | 26.7 | 63.3 | 18.6 | 64.1 | 13.5 | 63.3 | 9.75 | 62.4 | 7.72 | 62.1 | 7.21 |
| 1.78 | 45.6 | 24.1 | 62.3 | 14.4 | 62.1 | 9.94 | 61.0 | 7.02 | 60.1 | 5.49 | 59.8 | 5.12 |
| 2.00 | 46.4 | 21.7 | 61.0 | 10.9 | 60.2 | 7.27 | 59.0 | 5.03 | 58.1 | 3.90 | 57.8 | 3.63 |
| 2.24 | 47.1 | 19.6 | 59.5 | 8.16 | 58.4 | 5.27 | 57.3 | 3.59 | 56.5 | 2.77 | 56.3 | 2.57 |
| 2.51 | 47.7 | 17.6 | 58.0 | 6.05 | 56.9 | 3.81 | 55.8 | 2.56 | 55.2 | 1.96 | 55.0 | 1.82 |
| 2.82 | 48.1 | 15.8 | 56.7 | 4.44 | 55.6 | 2.74 | 54.7 | 1.82 | 54.1 | 1.39 | 54.0 | 1.29 |
| 3.16 | 48.5 | 14.1 | 55.5 | 3.24 | 54.5 | 1.96 | 53.7 | 1.29 | 53.3 | 0.98 | 53.2 | 0.91 |
| 3.55 | 48.8 | 12.6 | 54.5 | 2.35 | 53.6 | 1.40 | 53.0 | 0.92 | 52.6 | 0.70 | 52.5 | 0.65 |
| 3.98 | 49.0 | 11.3 | 53.7 | 1.69 | 52.9 | 1.00 | 52.4 | 0.65 | 52.1 | 0.49 | 52.0 | 0.46 |
| 4.47 | 49.2 | 10.1 | 53.0 | 1.22 | 52,3 | 0.71 | 51.9 | 0.46 | 51.6 | 0.35 | 51.6 | 0.32 |
| 5.01 | 49.4 | 9.02 | 52.4 | 0.87 | 51.9 | 0.51 | 51.5 | 0.33 | 51.3 | 0.25 | 51.3 | 0.23 |
| 5.62 | 49.5 | 8.06 | 51.9 | 0.62 | 51.5 | 0.36 | 51.2 | 0.23 | 51.0 | 0.18 | 51.0 | 0.16 |
| 6.31 | 49.6 | 7.19 | 51.5 | 0.45 | 51.2 | 0.26 | 51.0 | 0.16 | 50.8 | 0.12 | 50.8 | 0.11 |
| 7.08 | 49.7 | 6.41 | 51.2 | 0.32 | 50.9 | 0.18 | 50.8 | 0.12 | 50.7 | 0.09 | 50.6 | 0.08 |
| 7.94 | 49.8 | 5.72 | 51.0 | 0.23 | 50.8 | 0.13 | 50.6 | 0.08 | 50.5 | 0.06 | 50.5 | 0.06 |
| 8.91 | 49.8 | 5.10 | 50.8 | 0.16 | 50.6 | 0.09 | 50.5 | 0.06 | 50.4 | 0.04 | 50.4 | 0.04 |
| 10.0 | 49.8 | 4.55 | 50.6 | 0.11 | 50.5 | 0.06 | 50.4 | 0.04 | 50.3 | 0.03 | 50.3 | 0.03 |
| 11.2 | 49.9 | 4.06 | 50.5 | 0.08 | 50.4 | 0.05 | 50.3 | 0.03 | 50.3 | 0.02 | 50.3 | 0.02 |
| 12.6 | 49.9 | 3.61 | 50.4 | 0.06 | 50.3 | 0.03 | 50.2 | 0.02 | 50.2 | 0.02 | 50.2 | 0.01 |
| 14.1 | 49.9 | 3.22 | 50.3 | 0.04 | 50.2 | 0.02 | 50.2 | 0.01 | 50.2 | 0.01 | 50.2 | 0.01 |
| 15.8 | 49.9 | 2.87 | 50.3 | 0.03 | 50.2 | 0.02 | 50.2 | 0.01 | 50.1 | 0.01 | 50.1 | 0.01 |
| 17.8 | 50.0 | 2.56 | 50.2 | 0.02 | 50.2 | 0.01 | 50.1 | 0.01 | 50.1 | 0.01 | 50.1 | 0.01 |
| 20.0 | 50.0 | 2.28 | 50.2 | 0.01 | 50.1 | 0.01 | 50.1 | 0.01 | 50.1 | 0.00 | 50.1 | 0.00 |
| 22.4 | 50.0 | 2.04 | 50.1 | 0.01 | 50.1 | 0.01 | 50.1 | 0.00 | 50.1 | 0.00 | 50.1 | 0.00 |
| 25.1 | 50.0 | 1.81 | 50.1 | 0.01 | 50.1 | 0.00 | 50.1 | 0.00 | 50.1 | 0.00 | 50.1 | 0.00 |
| 28.2 | 50.0 | 1.61 | 50.1 | 0.01 | 50.1 | 0.00 | 50.0 | 0.00 | 50.0 | 0.00 | 50.0 | 0.00 |
| 31.6 | 50.0 | 1.44 | 50.1 | 0.00 | 50.1 | 0.00 | 50.0 | 0.00 | 50.0 | 0.00 | 50.0 | 0.00 |
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When a current transformer bridge with no LF compensation is
used to monitor the adjustment of an impedance matching network,
the result is a transmitter load impedance which is too low in
magnitude at low frequencies, a situation which is particularly
damaging for transistor power-amplifiers (see discussion of power-transfer
efficiency given in AC
theory, section 1-28). The situation when Herzog's
LF compensation is applied is an improvement, in that the load
impedance magnitude obtained by balancing the bridge rises as
the frequency is reduced (until the point where the falling reactance
of the transformer coil begins to dominate); and although the
bridge can not be considered to be accurate at low frequencies,
the error incurred is harmless in that it will merely cause a
small reduction in transmitter output power. Loading a transmitter
with 60W instead of 50W
will not be noticeable to the operator, and will have little
effect in terms of signal strength at the receiving station;
and so we may regard a magnitude error of +20% as acceptable.
This implies an SWR of 60/50=1.2 on any 50W
cable between the transmitter and the bridge, which is not serious.
The point with which we may take issue however, is that the resulting
bridge is inaccurate, wherease bridges compensated by modification
of the voltage sampling network are not. With regard to the phase performance (referring to fig. 3), we may observe that critical damping of the current transformer secondary network is very effective at bringing the load phase angle close to 0° over a wide frequency range. If we adopt the criterion recommended by Underhill and Lewis [2], that a phase error of 7° or less is inoffensive to the generator, we see that an uncompensated bridge with a 10mH transformer secondary and a 50W secondary load is unsatisfactory at frequencies below about 7MHz, whereas the maximally boosted Herzog bridge (Rh=50W, Ri omitted) gives acceptable performance down to about 1.7MHz. We may also note that the maximally boosted circuit gives the best compensation overall; an outcome which we might have expected given that Ri does not appear in the condition for critical damping (equation 2). It appears that the reason why Herzog used partial LF-boost (finite values for both Ri and Rh) in his patent [q.v.] is that he needed Ri to be present so that he could place a small inductor in series with it in order to compensate for the propagation delay or parasitic parallel secondary capacitance of the current transformer. The principal advantage of Herzog's LF compensation scheme is that it can be used when modification of voltage sampling network is either difficult or undesirable; and results in a bridge which, although inaccurate, behaves in a way which will not reduce the life-expectancy of any transistor power amplifier connected to it. Given that it is already possible to make accurate bridges by adjusting the voltage sampling network to have the same frequency dependence as the current sampling network, it appears to have little merit it in its intended form; but as is so often the case, what the inventor had in mind is not the reason why the circuit is interesting. Since no theoretical analysis was given in the original references, it is difficult to know how thoroughly the circuit had been investigated when the patent was written; but the simple view that the series capacitor reduces the loading on the current transformer at low frequencies does not do justice the actual circuit behaviour. When the compensating capacitor is chosen to give best phase accuracy, there is a substantial increase in current transformer output voltage as the frequency is reduced, which suggests that we might connect to the network in a different way in order to counteract this tendency. This leads to a circuit rearrangement, not covered by Herzog's patent, which useful for the construction of accurate RF ammeters, and is referred to in these documents as the maximally-flat current-transformer (see Appendix 6.2 and Appendix 6.3). |
© D W Knight 2007.
David Knight asserts the right to be recognised as the author of this work.
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